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Let A(G) be the adjacency matrix of G. The characteristic polynomial of the adjacency matrix A is called the characteristic polynomial of the graph G and is denoted by φ(G, λ) or simply φ(G). The spectrum of G consists of the roots (together with their multiplicities) λ 1(G) ? λ 2(G) ? … ? λ n (G) of the equation φ(G, λ) = 0. The largest root λ 1(G) is referred to as the spectral radius of G. A ?-shape is a tree with exactly two of its vertices having maximal degree 4. We will denote by G(l 1, l 2, … l 7) (l 1 ? 0, l i ? 1, i = 2, 3, …, 7) a ?-shape tree such that $G\left( {l_1 ,l_2 , \ldots l_7 } \right) - u - v = P_{l_1 } \cup P_{l_2 } \cup \ldots P_{l_7 }$ , where u and v are the vertices of degree 4. In this paper we prove that ${{3\sqrt 2 } \mathord{\left/ {\vphantom {{3\sqrt 2 } 2}} \right. \kern-0em} 2} < \lambda _1 \left( {G\left( {l_1 ,l_2 , \ldots l_7 } \right)} \right) < {5 \mathord{\left/ {\vphantom {5 2}} \right. \kern-0em} 2}$ .  相似文献
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A Π-shape tree is a tree with exactly two vertices having the maximum degree three. In this paper, we classify the Π-shape trees into two types, and complete the spectral characterization for one type. Exactly, we prove that all graphs of this type are determined by their Laplacian spectra with some exceptions. Moreover, we give some L-cospectral mates of some graphs for another type.  相似文献
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A subdivision vertex-edge corona G_1~S?(∪ G_3~E) is a graph that consists of S(G_1),|V(G_1)| copies of G_2 and |I(G_1)| copies of G_3 by joining the i-th vertex in V(G_1) to each vertex in the i-th copy of G_2 and i-th vertex of I(G_1) to each vertex in the i-th copy of G_3.In this paper, we determine the normalized Laplacian spectrum of G_1~S?(G_2~V∪ G_3~E) in terms of the corresponding normalized Laplacian spectra of three connected regular graphs G_1, G_2 and G_3. As applications, we construct some non-regular normalized Laplacian cospectral graphs. In addition, we also give the multiplicative degree-Kirchhoff index, the Kemeny's constant and the number of the spanning trees of G_1~S?(G_2~V∪ G_3~E) on three regular graphs.  相似文献
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A graph is called Laplacian integral if all its Laplacian eigenvalues are integers. In this paper, we give an edge subdividing theorem for Laplacian eigenvalues of a graph (Theorem 2.1) and characterize a class of k-cyclic graphs whose algebraic connectivity is less than one. Using these results, we determine all the Laplacian integral tricyclic graphs. Furthermore, we show that all the Laplacian integral tricyclic graphs are determined by their Laplacian spectra.  相似文献
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